UGH ‘. BED - - O R {Ty}: _ I ERS ...--, v-w .- .o-.< «my «3.qu vr-zfv-g . : NJ. , as; Ti TRANSFER 1.1M )- , . M if?! iMlCHl-GRN't‘STAIE .——-u--.n-.-h<.qq ~-.u “a, . feisls . , _,.....M.. _ 3%”: ill?“ ,./ ‘Q MyMWMMIIIIILIIJMM"mug/WI L I B R A R Y Michigan Sues University This is to certify that the thesis entitled MECHANISMS OF HEAT TRANSFER THROUGH ORGANIC POWDER IN A PACKED BED presented by Albert Chun—yung Chen has been accepted towards fulfillment of the requirements for £h'_D'___degree in. A515 Eng - 19. (l. 9thhm Major professor Date flerl/ 4 [€47 . j '-' ,.: ‘3; Al‘09 MECHANISMS OF HEAT TRANSFER THROUGH ORGANIC POWDER IN A PACKED BED By Albert Chun—yung Chen Due to an ever increasing demand for convenience— type foods, many food products are being produced in dehydrated and powdered formso Most of the dried and powdered products are subjected to either cooling or heating during processing. In order to effectively con— trol the temperature level and quality of the products, reliable information regarding the thermal properties of the powdered food is needed. The objectives of this research is to describe and represent quantitatively the transfer of heat through a packed bed of small organic powder particles, A mathematical model was derived to predict the effective thermal conductivity of a powder which contained solids with known thermal conductivity. The basic parameters considered were: thermal and mech— anical properties of dry milk solids, thermal conductivity of the interstitial gas, void fraction of the packed bed, particle size and size distribution, temperature level, moisture content and structure of the packed bed. In order to measure the thermal and mechanical properties of dry milk solid, regular nonfat dry milk Albert Chun-yung Chen was compressed by a hydraulic press until the bulk density of the compressed dry milk was equal to the density of dry milk solids (1.46 gm. per ml.). The thermal and mechanical properties of the specimen thus made was used to approxi- mate the thermal and mechanical properties of the particle solids. The thermal conductivity of the specimen was measured by a transcient thermal property measuring facility. The mechanical properties were measured by an Instron testing machine. The effective thermal conductivity of powdered milk was measured by a steady—state thermal conductivity measuring apparatus. The thermal conductivity of dry milk solids was found to depend on the temperature and moisture content of the solid. Its value was 0.2699 Btu/hr ft OF at 140 OF and 3.5% moisture content. The primary mechanism of heat trans- fer through organic powder in a packed bed was found to be conduction through the particle solids. In general, this contribution was 93.3% of the effective thermal conductivity. Conduction through the gas phase was “~8% while conduction through contact points was 1.9% of the effective thermal conductivity. Due to the small contribution of the gas phase, different types of interstitial gases did not have a significant influence on the values of effective thermal conductivity of a powder bed. Temperature, moisture Albert Chun—yung Chen content and void fraction were the dominating parameters for the values of effective thermal conductivity of a powder bed. Approved 9 K HAAVW Major Professor -avetlufi Department Chairman Date ; I MECHANISMS OF HEAT TRANSFER THROUGH ORGANIC POWDER IN A PACKED BED By Albert Chun—yung Chen A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1969 For their personal sacrifice, understanding and encourage- ment, this thesis is dedicated To my wife, Judy new—born son, Mark and parents, Mr. and Mrs. Ching Chuan Chen ii ACKNOWLEDGMENTS The author wishes to express his sincere apprecia— tion and gratitude to: Dr. D. R. Heldman, Assistant Professor, Agricul— tural Engineering Department and Food Science Department, for his constant inspiration, encouragement, interest and guidance. Dr. C. w. Hall, Chairman and Professor, Agricul— tural Engineering Department, for the graduate assistant— ship that enabled the author to undertake this inves— tigation; also for his interest and assistance in guiding this program. Dr. A. M. Dhanak, Professor, Mechanical Engineering Department, for his depth and thoroughness in teaching heat transfer at the graduate level and serving as a guid- ance committee member. Consultations were freely given and most rewarding. Dr. T. I. Hedrick, Professor, Food Science Depart— Inent, for consultations, using his laboratory and serving as a guidance committee member. Dr. A. W. Farrall, Professor, emeritus, Agricultural IEngineering Department, for opportunity to work on steady— S‘tate thermal conductivity measurement of various types OI“ dry milk. Dr. J. V. Beck, Associate Professor, Mechanical Engineering Department, for using his laboratory to mea— sure thermal conductivity of milk solid and for the use of his computer program which was partially supported by the National Science Foundation. Timothy w. Evans for help in obtaining data on the transient thermal properties measuring facility. Thomas G. Kamprath for help in running the test of contact number. Dr. B. A. Stout, Professor, Agricultural Engineering Department, for using his Instron testing machine to meas— ure mechanical properties of dry milk solid. iv TABLE OF CONTENTS ACKNOWLEDGMENTS . . . . . . . . . . LIST OF TABLES. LIST OF FIGURES . . . . . . . . LIST OF APPENDICES NOMENCLATURE Chapter I. II. III. IV. INTRODUCTION . . . . . . OBJECTIVES Justification. . Objectives of This Research LITERATURE SURVEY AND JUSTIFICATION The Conductivity of the Interstitial Gas. The Effect of Temperature Level. The Effect of Bulk Density . Thermal Conductivity of Solid Material and Contact Conductance. The Effect of Particle Size Void and Porosity The Effect of Particle Size Distribution: The Effect of Moisture Content . The Effect of Pressure and Mechanical Properties of the Solid Material. DEVELOPMENT OF THEORY. Assumption. . l. The Mechanism of Radiation Heat Transfer is Negligible. . . Free Convection is Negligible Heat Flow is Assumed One Dimensional . . U0“) xiii (Der-lr‘l—l Chapter A. The Particles are Assumed Spherical and Smooth. . 5. Bulk Density is the Only Criterion for Mechanical and Thermal Proper— ties of Solid. . . . . . . Structure of Random Bed . . 1. Definition of a Random Bed . 2. Relation Between the Volume and Area Void . 3. Particle Size and Size Distribu— tion. A The Distribution Function of the Sizes of the Particles Appearing on a Cross—section of a Random Bed. The Model of Heat Transfer Mechanism V. INSTRUMENTATION, EQUIPMENT AND EXPERIMENTAL PROCEDURES. Preparation of Specimens l. Adjustment of Bulk Density 2. Adjustment of Moisture Content Instron and Mechanical Properties of Particle Solid. Transient Thermal Properties Measurement Facility. Steady— State Thermal Conductivity Measuring Apparatus Number of Contact Points Between Particles . . . . . . . . . . VI. RESULTS AND DISCUSSION Pressure, Bulk Density, Void and Porosity. 1. Pressure and Bulk Density. . 2. Bulk Density, Void and Porosity. Mechanical Properties of Powder Particles Thermal Properties of Particle Solid Effective Thermal Conductivities of Dry Milk . . Number of Contact Points of Spheres in a Random Bed . . . . . . vi Page 27 27 3O 30 31 3A 37 1+3 43 AU 47 50 55 57 59 59 6A 65 73 79 81 Chapter VII. ANALYSIS . . . . . . . . . . Combined Effects of Temperature and Moisture Content on KS . Contact Conductance and Number of Contact Points Predicted on the Cross- Section of a Random Bed . . l. Contact Conductance of a Single Point 2. Frequency Distribution Function G(s) and the Predicted Number of Contact Points . Contact Conductance. Effect of Each Parameter on Ke . Influence of Component Thermal Con— ductivities on Effective Thermal Conductivity Using Dimensionless Groups . Comparison of Predicted and Experimental Thermal Conductivities . . . VIII. CONCLUSIONS IX. LIMITATION OF THE MATHEMATICAL MODEL DEVELOPED . . . . X. RECOMMENDATIONS FOR FUTURE WORK BIBLIOGRAPHY APPENDICES . . . . . . . . . . . Page 84 8A 87 87 89 93 101 101 104 106 107 108 115 A1. LIST OF TABLES Pressure Effect on Bulk Density. Modulus of Elasticity of Regular Nonfat Dry Milk (II 36) Modulus of Elasticity of Dry Milk Solid and Some Common Materials Results for Solid Thermal Conductivity, K of Regular Nonfat Dry Milk. . Effect of Temperature to KS Effective Thermal Conductivity of Regular Nonfat Dry Milk . . Number of Contact Points of Spheres in a Random Bed . . . . . . . Predicted Number of Contact Points on the Cross—Section of a Random Bed Per cent of K1, K2 and K3Q5 to Ke (II 76) AT of Each Surface Region in Fig. 5.8(a). viii S, Page 61 7O 71 75 76 81 83 92 98 Figure 2.1. .LT—lr—D‘UO [.4 LIST OF FIGURES One Dimensional Heat Flow through Particu— late System in a Packed Bed . . Constant Temperature Lines in Representative Sample of Spheres in Cubical Array as Obtained by Relaxation Solution Variation of Ratio of Local Heat Flux (per unit area) to Arrange Heat Flux Across Plane A—A in Fig. 3.1 for Various Values ofKSg/K Effect of 8;Particle Size Distribution on Ke, e and x . . . . . . . . 0' Two Spherical Particles in Contact. Powder Solids of Regular Nonfat Dry Milk. A Packed Bed for Eq. (4.2) Cumulative Particle— size Distribution of Regular and Instant Dry Milk by Coulter Counter Method. . . Particle—size Distribution of Regular Non— fat Dry Milk . . . . . . . . A Section of Random Bed Model of Heat Transfer Mechanisms Hydraulic Press Specimen for Measurement of Solid Thermal Conductivity . . . . . . . . Specimen Press Instron Testing Facility . . ix Page 13 13 18 22 29 29 33 33 35 38 A5 45 A6 A8 Figure Page 5.5. A Sketch of Instron System . . . . . . A9 5.6. Optimun Transient Thermal Properties Measure— ment Facility . . . . . . . . . . 51 5.7. Sketch of Optimun Transient Thermal Proper— ties Measurement Facility . . . . . . 52 5.8. Thermocouple Locations of Specimens for Measuring Thermal Properties . . . . . 5A 5.9. Steady— state Thermal Conductivity Measuring Apparatus . . . . . . . . 56 5.10. The Container for Testing Contact Points. . 58 6.1. Effect of Pressure on Bulk Density. . . . 62 6.2. Pressure vs. Bulk Density. . . . . . . 63 6.3. Void and Porosity on the Cross—section of Packed Bed . . . . . . . . . . . 66 6.A. Measurement of Mechanical Properties . . . 68 6.5. Effect of Bulk Density on Modulus of Elasticity . . . . . . 69 6.6. Temperature Profiles in Measuring Thermal Properties of Dry Milk Solid . . . . . 7A 6.7. Effect of Temperature on Thermal Con— ductivity of Milk Solid and Air . . . . 77 6.8. Temperature Effect on Thermal Conductivi— ties of Food Products and Other Material . 78 6.9. Effect of Moisture Content on Ke of Nonfat Dry Milk. . . . 80 7.1. Effects of Mean Particle Size and Standard Deviation of Size Distribution on Function G(s) . . . . . . . . . . 91 7.2. Effect of Temperature on the Component and Effective Thermal Conductivities. . . . 95 Figure 7.3. Effect of Moisture Content on the Component and Effective Thermal Conductivities Effect of Bulk Density on the Component and Effective Thermal Conductivities. Effect of Temperature and Moisture Content on Effective Thermal Conductivity Effect of the Thermal Conductivities of Interstitial Gases on the Effective Thermal Conductivity of Nonfat Dry Milk Correlation for Thermal Conductivity of Nonfat Dry Milk in Packed Bed. Comparison of Predicted and Experimental Ke for Nonfat Dry Milk with Air as Inter— stitial Gas. Contact Area and Contact Angle Example of Temperature Profiles for Trans— ient Thermal Properties Measurement. xi Page 96 97 99 100 102 103 128 13A Appendix I. II. III. IV. VI. VII. LIST OF APPENDICES Reduction of Equation (A.12) Computer Program for Integration of Equation (A8) . . . . . Computer Program for Equation (A8) Evaluation of B, y, ¢, and 6 of Equation (A.18) . . . . . . . . . . The Relation Between the Contact Number, Contact Area, Contact Angle and Particle Size . . . . . . . . . . . . Justification of Heat Loss from the Side Wall of Specimen in Figure 18(a) Formula Used for Calculating Coefficient of Correlation . . . . . . . Page 116 120 121 123 127 132 U33> (5' f(-) F(x) G(s) G(O+) PS NOMENCLATURE Radius of contact area between two particles, (in). Cross—sectional area of a packed bed, (ft)2. Properties of solid material. Distance between the geometrical center of a particle and the contact surface between par- ticles, ft. Constants defined in Eq. (A1). Constant in Eq. (3.A). Apparent particle density, g/ml. Modulus of Elasticity, psi. Porosity inside particles. Log — normal density function. Function of Probability frequency distribution function of particle sizes. The distribution of particle sizes as they appear on a cross—section of a random bed. Number of Contact Points on the cross—section of a random bed predicted by Eq. (A7). Heat Transfer coefficient for thermal radiation be— tween particles, Btu/(hr)(°F)(sq. ft). Bulk modulus, Psi. Distance between the geometrical center of a particle and an unit area on the cross—section, ft. Thermal conductivity of solid material (milk), Btu/(hr.)(°F)(ft). xiii K Thermal conductivity of intersticial gas, /(hr.) g <°F>. K Thermal contact conductance, Btu/(hr.)(°F)(ft). K Effective thermal conductivity of a powder bed, Btu/(hr.)(°F)(ft). K Thermal conductivity with pressure in Eq. (3.A), p Btu/(hr.)(°F)(ft). K Radiation contribution to the effective thermal conductivity of a packed bed, Btu/(hr.)(°F)(ft). Kl Contribution of solid phase on a cross—section to the effective thermal conductivity of a packed bed, Btu/(hr.)(°F)(ft). K2 Contribution of gasous phase on a cross—section to the effective thermal conductivity of a packed bed, Btu/(hr.)(°F)(ft). K345 Contribution ofa.contact point on a cross-section to the effective thermal conductivity of a packed bed, Btu/(hr.)(°F)(ft). L Length of a packed bed, ft. A1 Length of the unit cell in Fig. A.6(a), ft. 1 Effective path length (Fig. A.6) for solid particles, or, the thickness of a slab of solid material which would offer the same resistance to heat transfer as the spherical shaped particles, ft. 1 Effective path length (Fig. A.6) between adjacent g solid particles, or, the thickness of a slab of stationary fluid which would offer the same heat transfer resistance as the filaments of fluid near the contact points between particles, ft. M First moment of F(x), ft. IF MC Moisture Content, %. P Pressure applied, psi. q,qavg Heat flux 1n Fig. 3.2, Btu/(hr.). q Heat flux in a random packed bed. r Average radius of particles in a packed bed, ft. 0 S Particle size as they appear on the cross—section of a random bed, (Fig. A.5), ft. T Temperature AT, ATS, AT Temperature drop in unit cell, solid g phase and gas phase, respectively, °F, in Fig. A.6. uo,ul Radius of stagnation area and contact area, respectively, in Fig. A.6, ft. W Constant in Eq. (A.lO). w Loading, lbs., in Eq. (3.1). x Diameter of particles in a packed bed, ft. xo Average diameter of particles in a random bed, ft. Greek B,¢,y,6 Parameters in Eq. (A.18). e Void between particles. 8F Void and porosity on the cross—section of a packed bed. 52 Void on the cross—section of a packed bed for heat flux through the interstitial gas phase, Fig. 645 Void on the cross—section of a packed bed for heat flux through the stagnation fluid around the contact points, Fig. 1 Fraction of the cross—section for heat flux through solld phase, Fig. . :3 Fraction of the cross-section for heat flux through contact area of contact p01nts, Flg. A.6. u Micron, unit for particle size of powdered milk. 9 Contact angle in Fig. A.6, radians. a Standard deviation of particle size. w Solid angle in Appendix V. v Poission Ratio. p Bulk Density, g/ml. XV CHAPTER I INTRODUCTION Due to an ever increasing demand for convenience— l type foods, many food products are being produced in de— hydrated forms. For example, milk powder, egg powder, coffee, powdered tea, etc. are well known powdered foods; some special products like powdered honey, molasses, caramel and a large variety of fruit and vegetable pow— ders; tomato powder, orange powder, bean powder, pumpkin powder and apple sauce-—the list is endless. As the result of cooperative research by engineers and food scientists, the quality and quantity of these convenience—type foods has been highly improved for over a quarter of a century such that, in fact, powdered food is now one of the most important branches of the food industry. Most of the dried and powdered products are subjected to either heating or cooling during proc— essing. The quality of milk powder (solubility, bacteria count, flavor and appearance) is affected by the temperature of the product during manufacturing, packaging, and storage as well as the quality of milk. In order to effectively control the quality and tempera- ture level of the products, reliable information regard— ing the thermal properties of the powdered food is needed. However, there is only limited information available on this subject. tha et_al. (1966) have done some prelimi- nary measurements on the thermal conductivities of non— fat, regular, spray, milk powder and wheat flour. The effects of mean operating temperature and moisture content on the effective thermal conductivities of these powdered foods were determined. Farrall et al. (1968) continued this work on various types of milk powder. It was felt that the thermal conductivities of the powdered food might be affected by some other factors, such as bulk density, fat content, particle size and distribution and some geometric structures as well as moisture contents. Since the usefulness of experimental data is limited to conditions of measurement, it would be desirable to formulate some mathematical models to describe the influ— ence of these factors on the mechanism of heat flow through the powdered food, which is a particulate system. Particulate systems are studied in many areas of chemical engineering. Particles may be catalyst pellets, ion exchange beads, etc. (Beresford, 1967). Chemical engineers have been mainly concerned with particulate systems in connection with the design and the analysis of the performance of packed bed equipment, when the bed is used as a heat—exchange device and when it serves as a catalytic reactor. Obviously, the thermal conductivity of such a system is the property of the most concern. A system of powdered food in packed bed can be analyzed as a model of particulate systems. CHAPTER II OBJECTIVES Justification The thermal conductivities of inorganic granular media ham been investigated by Wilhelm et_al. (l9A8) experimentally on macroscopic basis. The results of Kazarian and Hall (1963) and tha et_al. (1966) for ther- mal properties of food grains were also investigated by macroscopic approach. Since the transfer of heat through granulated material is of considerable importance as discussed above, it therefore seems highly desirable to establish some models which can be used to analyze the measured effective thermal conductivities of granulated materials, or of any heterogeneous systems. In addition, with such models, it will also be possible to predict the effective conductiv— ity of a heterogeneous system from the conductivities of its constituent parts. It is evident, however, that the task of evolving a general theory for the conduction of heat through a hetero— geneous system is very complicated. The difficulty is due to its complex bed structure. Even in the comparatively simple case of the conduction of heat through a system of A uniform sphere packed in a regular way, the mathematical difficulties are such that the problem has been studied only briefly to date (Masamune and Smith, 1963). In fact, many naturally or artificially granular materials have a random void structure. Where the system consists of granulated material of undefined shape and where the packing is irregular with a particle size distribution, a strictly mathematical analysis is entirely impossible. This fact, however, need not be discouraging entirely, be- cause it may still be possible to obtain a mathematical formula, arrived at by unorthodox methods based on sta— tistical probability theory for the bed structure, which might have sufficient accuracy and considerable value in practice. Objectives of This Research The overall objectives of this research are to des— cribe and represent quantitatively the transfer of heat through a packing bed of small powder particles (Figure 2.1), and derive mathematical models to estimate the effective thermal conductivity of a powder, K in Eq. (2.1), e made from a solid with known thermal conductivity. AT AX (2'1) q = -Ke A The primary parameters involved would be as follows: 1. The thermal conductivity of the solid material. r—:"l'l'_—‘ He at Eource <2: . ..Q A Sink \\\\\\\\\\\\\\\\\\\\\ o \\\\\\\\\\\\ \\\\\\\\ Heat Figure 2.l.——One Dimensional Heat Flow through Particulate System in a Packed Bed. 9 . 10. The thermal conductivity of the intersticial gas between the granular particles. The extent of void between the particles and, sometimes,the porosity within each particle. The bulk density of the powder bed. The mean particle size. The particle size distribution. The content number of a sphere in a random bed. Mechanical properties of the solid material, such as modulus of elasticity and Poisson's ratio. Temperature level of measuring. Moisture contents. Knowing these parameters for a heterogeneous pow— dered bed, the general mathematical models, as the result of this research, should be able to predict the effective thermal conductivity of the bed and the effect of each parameter for various types of powder, physical structure and processing conditions. The thermal properties as well as the qualities of the powdered food product could be controlled and improved by altering processing condi- tions. CHAPTER III LITERATURE SURVEY AND JUSTIFICATION Scarlett (1967) presented a relatively complete review of the literature on the topic of thermal conduc— tivity of inorganic powder. The study of thermal conductivity of a heterogen— eous medium was started by Maxwell (1881). It was found that, at atmospheric pressure, the conductivity of the powder was greater than that of the air. Smoluchow— ski (1910) conducted a systematic study on the thermal conductivities of several powders at various air pressures from 0.2 to 760 mm Hg. to investigate the effect of the air pressure on the thermal conductivity. He plotted the effective thermal conductivity value against the logarithm of the interstitial gas pressure and found that it was a S—shaped curve. He concluded that the conductivity of the powder depends mainly on pressure and nature of the inter— stitial gas. Some later investigation agreed with Smoluchowski's conclusion in general, but several authors did not agree and stated that gaseous phase is not the only factor in control. No theory was given to relate the conductivity of a heterogeneous system to the conductivi— ties of its constituents satisfactorily. Some new 8 approaches were suggested. The importance of investi— gating the effect of each parameter on the heat transfer instead of overall changes due to several parameters was stressed (Rowley et_a1., 1951). The following literature review is grouped for each parameter involved. The Conductivity of the Interstitial Gas The importance of the insterstitial gas (called K8; in the following chapters) was first observed in 1910 by Smoluchowski as mentioned above. Kannuluck and Martin (1933) measured the conductivities of some inorganic powders when the powders were filled with helium, which has a con- ductivity higher than that of the air. As was expected, they found that the conductivity of the powder was higher than when this product was filled with air. Schuman and Voss (193A) observed the same response with hydrogen. Meanwhile, Kistler and Caldwell (193A) filled the silica aerogel powder with dichloro—difluoromethane, which has a lower conductivity, and found that the conductivity of the powder was lower than that with air, as was expected. Prins et_a1. (1951) confirmed the results of earlier workers that the effective thermal conductivity of a pow— der would be increased with a interstial gas of higher conductivity. Recently Deissler and Boegli (1958) con— ducted an experimental study to determine the effective thermal conductivities of magnesium oxide, stainless 10 steel, and uranium oxide powders in various gases. They concluded that the effective conductivity of a powder is a strong function of the conductivity of the interstitial gas. The Effect of Temperature Level The effect of temperature level on the effective conductivity of a powder is two fold: the thermal con— ductivities of the constituents and the contribution of radiation. An organic powder could be considered as a solid- gas system or a mixture of amorphous materials. ”If a mixture of amorphous materials forms a heterogeneous system, its thermal conductivity may be considered in first approximation as an additive property and calculated by summation of the contributions of constituents" (Kowal— czyk, 195A). Furthermore, "Since K of both constituents increases with temperature, the thermal conductivity of a solid—gas system always increases with temperature" (Perry, 1950; Kowalczyk, 195A). A similar conclusion has been obtained experimentally (Deissler and Boegli, 1958; Kazarian and Hall, 1963). As the temperature is increased, the radiation from solid to solid through the interstices plays an increas— ing part of overall heat transfer (Russe11,1935; Laubnitz, 1959; Bretsznajder and Ziotkowski,l959). However, 11 Chapter IV shows the contribution of radiation to be nearly negligible for the practical situation of organic powders. The Effect of Bulk Density A packed bed of organic powder could be classified as a two phase system of solid and gas. ”The thermal conductivity of a solid—air system at constant tempera— ture is a function of apparent density" (Perry, 1950). ”This proves that the thermal conductivity of hetero- geneous system is an additive property" (Kowalczyk, 195A). Farrall et_a1. (1968) conducted an investigation on the effect of bulk density on the thermal conductivity of various types of dry milk. The type of samples in— cluded non—fat and whole dry milk prepared by regular spray, roller spray and foam spray processes. An interest— ing result that the thermal conductivity increased line— arly with the bulk density was observed for each type of powdered milk. The bulk density in turn is influenced by spray drying conditions (Hayashi et_al., 1967a). In general, bulk density increased with increasing solids content, preheat treatment, and pump pressure and decreasing in— let air temperature. 12 Thermal Conductivity of Solid Material and Contact Conductance The powdered material can be characterized if the nature of the solid and the particle size distribution of the powder are known. The conductivity of the solid grains influences the conductivity of the powder, but it would seem to be a secondary effect (Scarlett, 1967). Deissler and Boegli (1958) conducted an experi— mental study to determine the effective thermal conduc— tivity of magnesium oxide, stainless steel, and uranium oxide powder in various gases including the air. They found that the effective conductivity of a given void in a powdered system is greatly influenced by the arrangement of the material for high values of KS/Kg, whereas for low value of Ks/Kg the arrangement of the material is of lesser importance. For a given value of Kg (air for example), the value of conductivity of the solid material thus determined the importance of the arrangement of the particulate system. This could be visualized from the characteristic of small regions or points of contact between the particles in the powder. For higher values of KS/Kg such as the case of metal powder, most heat flow takes place in the vicinity of these points of contacts because the gas acts as an insulator at points where the particles are separated. Figure 3.1 shows the calculated temperature distribution 13 HEAT FLOW a “AVE. LOCAL HEAT FLUX I AVERAGE HEAT FLUX (a) k../k.,, 3 HEAT FLOW DISTANCE FROM POINT OF CONTACT SPHERE RADIUS Figure 3.2.-—Variation of Ratio of Local Heat Flux (per unit area) to Average Heat Flux Across Plane A-A in Fig. 3.1 for Various Values of k /k (Deissler and Boegli,s 1958). (b) k./k,, 30 Figure 3.l.——Constant Temperature Lines in Representative Sample of Spheres in Cubical Array as Ob— tained by Relaxation Solution (Deissler and Boegli, 1958). 1A for heat flow through spheres in cubical array. As the Ks/Kg increases from 3 to 30, the contact temperature lines crowd together in the vicinity of the point of contact; that is, more of the heat flow across the gas spaces takes place near the point of contact. It is evi— dent that for values of Ks/Kg on the order of 1000 nearly all the heat flow will take place through an extremely small area near the point of contact (Deissler and Boegli, 1958). This effect can be seen more clearly in Figure 3.2, where ratio of local heat flux to average heat flux across the plane containing the point of con— tact (plane A—A in Figure 3.1) is plotted against dis— tance from the point of contact divided by sphere radius. Values of q/qavg. on the order of 5000 were indicated near the point of contact for high values of Ks/Kg' There— fore, for high values of KS/Kg,the effective conductivity will be very sensitive to the exact way in which the particles make contact and to slight irregularities on the surface near the points of contact (Deissler and Boegli, 1958). However, for organic powder (nonfat dry milk) the Ks/Kg value is comparatively low so that the contact conductance, the arrangement of particles and the regularity of particle surface would not be as important as in the case of metal powder. 15 The Effect of Particle Size The influence of particle size is two fold. First of all, it affects the bulk density. Secondly, it af— fects the mechanism of heat transfer. Hayashi et_al. (1967b) found that the bulk density of nonfat regular spray dry milk increased gradually as particle size was decreased. This result is as expected since interstice between larger product particles will be larger and will reduce the bulk density. Duffie and Marshall (1953) also indicated that small particles may be inherently more dense than larger ones dried under the same condition. For roller dried skim milk, it was found that its density varied from 0.3 to 0.5, depending primarily on the fineness of grinding (Whittier and Webb, 1950, Coulter et_al., 1951). The increasing bulk density with smaller particles could result in higher thermal conductivity of the powder. For other inorganic mate— rials, it was found that the thermal conductivity of a glass spheres—air system increased while particle size decreased (Schotte, 1960). However, for metallic powder where K8 is much higher than Kg and thus K8 is the maxi— mum contribution to Ke, a fine powder will have a lower conductivity than a coarse powder packed to the same porosity because of the larger number of gas—solid inter— faces which acts as the resistance in the heat path (Scarlett, 1967). This would not be the case for dry 16 milk powder, because Ks of organic powder would not be as high as that of metallic powder. The second effect of particle size is on the mechan— ism of heat transfer. Waddams (19AA) measured the conduc— tivity of steel spheres and coarse calcite powder at atmos- pheric pressure. He found that the obnductivity increased with particle size, and the result was contributed to the mechanism of convection heat transfer as the particle size was increased. Kennuluck and Martin (1933) had similar observations for carborundum powder, magnesium oxide, glass and diphenylamine powders. However, the particle size of the organic powder is not large enough to consider convection. More details will be discussed in Chapter IV. Void and Porosity Generally, the bulk density of dry milk includes two kinds of air when packed: porosity and void. Por— osity is the air contained in each unit particle, while void is defined as the space contained between each unit particle in a container (Hayashi §t_al., 1967b). However, void defined above was usually called porosity in most works where there is no air inside each unit particle. The dependence of conductivity on porosity can be explained very easily; the greater the proportion of the bed occupied by gas spaces, the higher the resistance of the bed (Scarlett, 1967). This could be visualized as 17 Kg is less than that of Ks' Marathe and Tendolkar (1953) measured the conductivities of marble haematite and copper powder and concluded that conductivity varied linearly with the porosity of the bed. The Effect of Particle Size Distribution The influence of the particle size distribution of a powder is not clear. Scarlett (1967) was the first to investigate this problem. He measured the conductivi— ties of aluminum powder for various sizes and mixtures of different sizes. He concluded that the conductivity is a function of both the porosity and the mean particle size and; since the porosity depends in turn on the particle size distribution, the conductivity is a function of both mean particle size and the size distribution. Having made the conclusion above, he demonstrated qualitatively by mixing the coarse particles of less than 150 mesh and the fine particles of greater than 300 mesh grades of powder together in different proportion to pro— duce a range of porosities and mean sizes. Figure 3.3 shows the thermal conductivity of the powders at atmos— pheric pressure plotted against the percentage of the fine particles in the mixture. The porosity of the packed pow— der and the mean particle size are shown. The porosity decreased initially as the fine particles filled the pore spaces between the large particles and passed a minimum value at A0% by weight of fines. However, the maximum l8 ox Una w.ox so soapdnfihpmflm oNHm oHOthmm mo poo%%MI|.m.m oszmflm opmnw Emma oom swap mwoa & OOH om ow ow om om 0: cm om OH o _ _ _ _ a ___ ISO! I. I.o:. .I I. I.::. I rt ‘OX 9 om :N mm mm o: 2: ‘SM 00 mo OeS/SGIHOf l9 conductivity occurs for the 20% mixture and this is clearly due to the Opposing influence of the decreasing porosity and decreasing mean particle size. This experi— ment thus confirmed that a decrease in mean particle size can cause a substantial decreasing in effective con— ductivity. The above result was for aluminum powder which has a high KS. For organic powder which has a much smaller Ks’ the effect of mean particle size and porosity (or particle size distribution) on effective thermal con— ductivity might be different from Scarlett's result. The Effect of Moisture Content The fact that the thermal conductivity of a par— ticulate system increased with its moisture content has been reported (Kazarian and Hall, 1963; tha et_al., 1966; Farrall et_al., 1968). This could be due to two reasons. First of all, moisture content is dispersed in dry milk solid (Hall and Hedrick, 1966; King, 1965; Hayashi et_al., 1967a). It has been found that the thermal conductivity of solids increase with increasing moisture content (Patten, 1909). This is understandable since the conductivity of a liquid is much greater than that of a gas. It also could be visualized from the fact that a system in which the con- tinuous medium is a liquid will have a higher conduc— tivity (Scarlett, 1967). The second reason for Ke value 20 to be increased with moisture content might be due to the increasing of bulk density. Hayashi gt_al. (1967b) found that bulk density of nonfat regular spray dry milk increased gradually with its moisture content up to 6%. After moisture content of non—fat dry milk is greater than 6%, some physical changes would be expected, such as lactose crystallization or clustering of par- ticles (Hayashi et_a1., 1967a). The Effect of Pressure and Mechanical Properties of the Solid Material As the bulk density of a powder bed is increased either by application of pressure or by settling, void will be decreased. Meanwhile, the contact area and the number of contact points between particles will be af— fected. Mohsenin (1966) suggested that the following equa— tion be used to calculate the radius of the contact area between two particles under pressure: (Figure 3.A) _ wB 1/3 a — 0.721 —:T_]-_—— (3.1) X1 2 where w = Loading, lbs. xl, x2 = Diameter of two particles in contact, in. a = Radius of the contact area, in. 1 — 012 + l — v22 B= E E (3.2) 1 2 21 if where, E = Modulus of elasticity, psi v = Poisson's ratio. Thus the radius of contact area of two identical spheres in contact could thus be expressed as: 2‘l/3 Wx(1—\)) a = 0.721 ———9~—E—-———— (3-3) Since the contact thermal conductivity, or sometimes called "Residual Thermal Conductivities," is related to the contact area between two particles, the effect of pressure and mechanical properties of the solid material on Ke could thus be visualized. Also, according to Kowalczyk (195A): . . Pressure exerted on amorphous solids increases the contact area between molecules and atoms and should, therefore, promote heat conduction by atomic vibrations. This conclusion was confirmed by experiments of Bridgman (19A9) who has shown that for nonmetallic, amorphous solids (Pyrex glass, limestones, talc, etc.) the thermal conductivity increases proportionally to the pressure, according to the formula: 22 K = K + d p 1 atm 1000 (3.A) where p is the pressure in kilogram/cm2 and d is a numerical constant, whose value was determined experimentally for each material. X2 Figure 3.A.-—Two Spherical Particles in Contact. CHAPTER IV DEVELOPMENT OF THEORY Assumption Theoretically, it should be possible to calculate the effective conductivity of a given powder from the conductivities of the solid and gas by using Laplace's heat conduction equation in the solid and in the gas together with the boundary conditions at the interface. The actual calculation of the results by this method, however, appears to be impracticable because of the irregular shape and arrangement of the powder particles. (Deissler and Boegli, 1958.) Also, Cetinkale and Fishenden (1951) claimed that direct analytical solution proved impracticable because of the many boundary conditions to be satisfied. Therefore, some reasonable assumptions for the practical situation are necessary in order to make the formulation of the theory possible. There are five assumptions for the case of organic powder: 1. The Mechanism of Radiation Heat Transfer is Negligible The condition of radiation heat transfer was first pointed out by Smoluchowski (1910). Radiation may con— tribute significantly at high temperature, particularly when the particles are larger (Schotte, 1960; Wilhelm, at al., 19A8; Strong et al., 1960). Russell (1935) and 23 2A Damkdhler (1937) postulated some formula for calculating the contribution to heat transfer by radiation. However, the mean temperature range on which Laubnitz (1959) measured the conductivities was 100°C to 1000°C. Schotte (1960) pointed out that radiation would not be important unless the particle size is greater than 1nm1at tempera- tures above AOOOC, or for OJImm particles above 1500°C. For this reason the radiation was neglected in most research to date (Masamume and Smith, 1963; Wilhelm et_a1., 19A8). The upper limits for Wilhelm et_a1. (l9A8) to neglect radiation were 3—A1mnparticle size, and 8—10 atmospheres, 300°C. Therefore, the particle size and temperature level of the organic powder were practically within the range where radiation is negligible. Argo and Smith (1953) simplified Damkéehler's expres— sion for radiation contribution and suggested the following empirical equation: 3 _ e . Ta Kl” - Ll (Tr-E) XO (0.173) (W) (14.1) For example, non-fat, regular spray milk powder will give the following results: Assume void 8 = 0.57, mean particle size x0 = 55A = 0.18 x 10‘3 ft (A.2) 25 average temperature = 150°F = 610°R (A.3) then, by Equation (A.l), 3 _ 0.57 -3 610 Kr - “(gt—0757) (0.18 x 10 > (0.173) (W) (A.A) = 0.1125 x 10‘3 Btu/hr ft °F which is practically negligible as compared to the effective thermal conductivities, Ke, which ranges from 0.1 to 0.3 Btu/hr ft oF (see Table 6.6). 2. Free Convection_is Negligible As indicated in Chapter III when discussing particle size, convective heat transfer starts to play a role as the particle size of a particulate system becomes larger. However, convection cannot be an important mechanism of heat transfer through fine powders (Schuman and Voss, 193A; Russell,l935). Deissler and Eian (1952), Deissler and Boegli (1958), Schotte (1960) confirmed this result and indicated that free convection was not an important factor in determining the effective thermal conductivity of a powder at high pressures up to 100 atm; the thermal conductivity being independent of gas pressure. If there were approciable free convection in the powder, the conductivity would continue to increase with 26 gas pressure inasmuch as free convection is a function of the density of the gas. Wilhelm et_a1. (l9A8) indicated the limits within which free convection can be neglected by stating "Heat transfer is almost purely conductive provided the particle size, the gas pressure, or the temperature are not too high. Rough upper limits are 3—A mm diameter, 8—10 atmo— sphere, and 300°C." Russell (1935) also pointed out that with pore space greater than l/A in., convection occurs. For organic powders, the particle size and temperature are certainly within this limit of negligible free con- vection. 3. Heat Flow is Assumed One Dimensional Due to its complexity, most of the theoretical pre— dictions of effective conductivity have been derived assuming plane isothermal or parallel lines of heat flow (Russell, 1935; Gemant, 1950; Deissler and Eian, 1952; Woodside, 1958; Deissler and Boegli, 1958; Masamune and Smith, 1963). All of these previous investigations have one factor in common, that is, they assumed a regular cubical array for the arrangement of uniform spheres to simplify the random arrangement and size distribution in the particle situation. Thus,the results can only be order of magnitude estimates. L___ 27 In order to investigate the validity of one dimen- sional heat flow, Deissler and Eian (1952) applied relaxa— tion method on several regular arrangements of cylinder packing to investigate the effect of bending heat-flow lines. However, they did not improve their prediction of effective thermal conductivities because the relaxation method still could not account for irregularity of particle arrangement. Therefore, they accepted this assumption of isothermal plane because it gave better agreement between the theoretical and the experimental results. A. The Particles are Assumed Spherical and Smooth Obviously the shape and surface conditions of the particles do influence the effective thermal conductivity. By Scarlett (1967), polishing the spheres decreased the effective conductivity, but he did not offer any explanation. "Spray dried particles are usually spheri- cally shaped, but some may be elongated and generally range in diameter from 10 to 250p.” Also, ”The surface of the spray dried milk particles is usually smooth" (Hall and Hedrick, 1966; Hayashi, 1962). Therefore, this assumption is made to simplify the complex situation. 5. Bulk Density is the Only Criterion for Mechanical and Thermal Properties of Solid In order to investigate the mechanical and thermal properties of powder solid, the powder will be compressed 28 until the bulk density of the powder reaches the density of the powder solids. Figure A.l(a) shows a cross—section of solid material and Figure A.l(b) illustrates compressed powder until p is equal to p of solid. It is assumed here that the mechanical and thermal properties of Figure A.l(a) can be approximated by that of Figure A.l(b). Since the particle size of organic powder is so small, it is impossible to measure the mechanical and thermal properties of particle solid directly. The above assumption suggests that a compressed solid block of pow— der with the same bulk density as particle solids would approximate the mechanical and thermal properties of particle solids. Structure of Random Bed In practice we are generally concerned with the behavior of spheres or irregularly shaped particles when packed at random. The arrangement of such a packing is very difficult to determine, especially for particles with size distribution. No comprehensive relationships between the variables describing such a packing arrange- ment has been developed to date. However, one possible approach to such a problem has been shown by means of statistical methods (Debbas and Rumpf, 1966). 29 ///// (a) Powder solids (b) Compressed powder (p = 1.146 g/ml) SOlids (p = l.A6 g/ml) Figure A.l.--Powder Solids of Regular Nonfat Dry Milk r—HI Figure A.2.——A Packed Bed for Eq. (A.2). 3O 1. Definition of a Random Bed A random packing can be defined analogous to a random mixture in the following manner: "Every particle has the same probability to occupy each unit volume throughout the packed bed, and with regard to the orienta- tion every direction has the same probability" (Leschon- ski, 1967). From this definition, the following two relations could be deduced. 2. Relation Between the Volume and Area Void The volume void, 8, is defined as the ratio of the volume of void to the total volume of the packed bed including void and particles. The area void, EF, is analogously defined as the ratio of the area of the sec— tioned voids in a cross-section of the packing, to the total area of the sectioned voids and particles. From the definition of randomness, no cross-section ‘is preferred to the other, hence: (LI-5) i.e., the mean area void of all cross-sections is equal. Since 8F = constant, therefore d y _ (A.6) 31 (see Figure A.2) i.e., the mean volume void is equal to the mean area void in a random bed (Masamune and Smith, 1963; Rumpf, 1958). 3. Particle Size and Size Distribution Particle—size distribution studies are becoming increasingly important in numerous areas of agricultural engineering research. The particle size and size dis- tribution of dry milk is a basic and important physical property influencing product reconstitutability, packing density and dustability. Hayashi gt_al. (1967c) conducted some investigations on the size distribution of nonfat dry milk. Their results could be properly described by log— normal distribution as following: 1 1n x — 1n x 2 f(x) = —— exp {- 1/2 [:ng } (11.7) ¢2v 0g 8 where f(x) = log—normal density function, Eg = geometric mean of particle sizes = x0 cg = geometric standard deviation. When a log-normal distribution may be approximately assumed, the cumulative frequency data may be fitted as a straight line on log—probability paper. The abscissa value corresponding to 50 per cent level is the geometric mean and the ratio of the 8A.l3 per cent to the 50 per cent values is the geometric standard deviation. (Cooke and Bowen, 1966; Orr and DallaValle, 1959.) 32 Figure A.3 shows the log-normal distribution of regular nonfat dry milk plotted from the observation of Hayashi gt_al. (1967c). Also, the results of Mori (196A) for instant dry milk is shown. The abscissa values correspond to 50 per cent and 8A.l3 per cent are A5.5 micron and 76 micron, respectively. Therefore, for regular nonfat dry milk, X = A5.5 micron o = 76/A5.5 = 1.67 (A.8) Equation (A.7) along with Equation (A.8) gives the fre- quency distribution of particle sizes of regular nonfat dry milk, or f(x) in Figure A.A The area under curve in Figure A.A was found to be 50.1. By probability theory in statistics, the area under the curve should be equal to unity for the total range of particle sizes (Parzen, 1965). Then one unit on the ordinate scale will be determined as following: 1 _ - m ' 0-00196 ‘ 0:002 (“'9) Therefore, _ 0.002 F(x) — 0'1 f(x) 33 .5 ”' [3 O 280 micron aperature 100 micron aperature CI Instant nonfat dry milk Cumulative number per cent lllLI l I I I I I 9 10 20 30 A0 50 6070 &)100 ia meter, micron Figure .3 —-Cumulative Particle— sizne Distribution of Regular and Instant Dry Milk by Coulter Counter Math 200 0.020 0.018 0.016 0.01A 0.012 0.010 0.008 0.006 0.00A 0.002 IIIIllllllIIII 10 20 30 A0 50 60 70 80 90 100 120 1A0 160 Particle size, x, micron Figure A.A.--Part1cle—size Distribution of Regular Nonfat Dry Milk. 3A = w ——1—- exp {-1/2 [in—lfi—gfi—Xfif} (4.10) V2t 1n 0g g where _ 1 W ' at and F(x) is the probability frequency distribution func— tion of particle sizes of nonfat dry milk, with the follow- ing conditions: F(x) 2 0 and for all x (A.ll) A. The Distribution Function of the Sizes of the Particles Appear- ing on a Cross—section of a Random Bed Figure A.5(a) shows a section of a granular bed. The particles in the bed have a size distribution function F(x). The diameters, s, of the circles appearing on the cross—section of the granular bed with particles size dis— tribution F(x)have a frequency distribution function G(s) (Figure A.5[b]). The relation between F(x) and G(s) was suggested as follows 35 (a) A representative section of particulate system in a random bed W F(x), e 41 I (c) Relation between x and S (b) Cross-section AA (Area = A = 1) Figure A.5.——A Section of Random Bed. 36 G(S) = Mi Jx=xmax —2F—(X)2—r dx (A.12) 1F x=s (x - s )1 by consideration of probability (Debbas, 1966), where x M = max x F(x) dx (u.13) 1F 0 is called the first moment of F(x). Notice that in Equation (A.12) the limits of integration are from s to xmax indicating that any sphere with size smaller than s will not contribute to the frequency distribution function G(s) on the cross-section. Equation (A.12) could be reduced to the equation as follows for numerical integration: X 2 X max 0 s 1 0 1n — 1 2 G(s)=I-VIl—[%(X2-S) e O 1F S X=Xmax (A7) - g(X) dXJ X=S where Cl and C2 are constants for a given particle size distribution. Appendix I gives a complete derivation and definitions of Equation (A7). There may be two applications for Equation (A.12). Since Equation (A.12) gives the distribution of sizes of 37 sectioned particles on the cross-section surface, the inte- gration of Equation (A.12) should give the total sectioned area of solid. In addition, as s becomes very small, for example, 0.01u, G(0.01u) in Equation (A.12) would then possibly give the number of contact points on the cross— section surface A—A of Figure A.5. The Model of Heat Transfer Mechanisms Due to the complexity of geometry involved, one dimen- sional heat flow in a powder bed has been assumed. Under this assumption, any cross section of a random bed coulg be considered as an isothermal plane. Heat transfer through the isothermal plane (Section AA, Figure A.5(a)) could be described by the following mechanisms. Mechanism 1: Heat Transfer through the Solid Phase. Mechanism 2: Heat Transfer through the Interstitial Gas by Conduction and Radiation. Mechanism 3: Heat Transfer through the contact area of the contact points on the cross- section. Mechanism A5: Heat Transfer through the stagnation fluid around the contact points by conduction and radiation. These four mechanisms are sketched in Figure A.6(a). The effective area for each mechanism are denoted by e1, e2, e3 and 845 respectively. They can be evaluated as follows: 11/ 1A {a} I—l-HI— ———>+ 38 Pha ‘3 //////// \\ /:://4/ \. \\\ ////¢ that- Z’l‘idl on //// Gas Phaa e le—el—elleesele— 22—11" the I3 HO easectl o (b) Unit cell I':7.:-;ure h.0.~~Modcl of wIel of IIe t Transfer Meche (f) a 12) his 'ms 'fllrough We .391 ll WV/ TT ATS 3 3 I/1 / // 4 55/391, __;L 50*) G(o+) (c) Three—mechliism '7. l. m 17339., J. Tan-“'3 fer Mechan :3: +5HTT Model 39 6 =l—€-e (4.14) a = y e (4.15) 63 = ($3 + 645) 6 = €(l — y) 6 (4.16) 545 = (6 — 82) (l — 5) = €(l — y) (l — 6) (4.17) where ¢ = lg/Al B = Al/xO y = 62/6 6 = 53/(83 + 845) (4.18) The parameters ¢, 8, y and6 are evaluated in Appendix IV° Since four mechanisms operate in a parallel fashion, their separate contribution may be added to obtain the total heat flow across the cross—section. AT AT AT AT (H.19) 40 where >1 J: I 5 - Thermal conductivity of the series mechanism which represents heat flow through the solid phase and the stagnation fluid whose total length is Al. K145 = l l resistence resistence through + through stagnation solid phase _ l _ i5 13 K + 'K“ g s = l (14.20) ¢Al + (l — ¢) Al K K g s l KNS = ———————-—-—- (4.21) A + l _ d) Kg KS Therefore Equation (4.19) can be written as: Ke = (l — 6) KS + ye Kg + e(l — y)6 KS 1 + 6(1 — y) (l - 6) ————————————— (4.22) .9 + £.:.$. K K 41 In Equation (4.22) the first term is the contribution of the solid phase, the second term is the contribution of the interstitial gas and the last two terms are the con— tribution of the contact point. Let Kl = (l — 8) KS (4.23) K2 = ye Kg K = (l—€)6K + 5(1" )(1‘5) ° 3 L+l_-.ql K K g s Then Equation (4.22) is reduced to the following: K=K+K+K (4.24) e l 2 c K0 in Equation (4.24) represents the contribution of total contact points on the cross—section. If G(O+) represents the number of contact points expected on the cross—section of a random bed, then the contribution of a single contact point to Ke will be as following: = 4. K3“5 Kc/G(O+) ( 25) Thus Ke = Kl + K2 + K345 . G(o+) (4.26) 42 where Kl can be calculated from Equation (4.23), K2 can be calculated by Equations (4.23) and (A21), K345 can be cal— culated by Equations (4.25), (4.23), A18), (A31), A21), (A22), (A7), (A8), (A9) and (al). The value of KS for regular nonfat dry milk solids will be determined experimentally in the following chapters. The values of Kg for various types of gases could be found. For the case of air, Kg is effected by temperature as follows (Kreith, 1964): K8 = 0.0133 + 0.000021 T (4.27) CHAPTER V INSTRUMENTATION, EQUIPMENT AND EXPERIMENTAL PROCEDURES There are several parameters in the mathematical models shown in Chapter IV which must be determined experi— mentally. They are the mechanical properties of powder particles, thermal conductivities of dry milk solid, num— ber of contact points between particles in a packed bed and effective thermal conductivities of various powdered milk at various bulk densities and moisture contents. Preparation of Specimens l. Adjustment of Bulk Density Since the size of dry milk particles is so small (usually from 10 to 250p) the direct measurement of mechanical properties and thermal properties of particle solid is practically impossible. In order to prepare specimens of milk solids with workable dimensions, the bulk powdered milk was compressed mechanically until the bulk density of the specimen approached the density of particle solid. Then the mechanical and thermal proper— ties of the specimen were measured to approximate the mechanical and thermal properties of the dry milk solids. 43 44 The powdered milk was compressed manually with a hydraulic press1 shown in Figure 5.1. The maximum force available from the press is 60 tons. The cylinder and piston for making specimens are sketched in Figure 5.3(a). The cylinder, C in Figure 5.3(c), was made of steel AISI 4340 and the thickness was designed so that there was no appreciable expansion of the cylinder as the pressure of the piston reached as high as 70,000 psi (Lampi §t_a1., 1965). The sliding collar was installed in order to eliminate the friction between the specimen and the cylin— der wall. After the bulk density of the specimen approached the density of particle solid (1.44 to 1.48 g/ml; Hall and Hedrick, 1966), the specimen was removed from the cylinder by continuous application of hydraulic pressure. Speci— mens made by the 3 in—cylinder shown in Figure 5.3(b) were required for the transient thermal properties measurement facility. Figure 5.3(0) shows the tube used to prepare square specimens. Figure 5.2 shows the specimens made. 2. Adjustment of Moisture Content The moisture content of commercial nonfat dry milk ranges from 3 to 4%. In order to investigate the effect of moisture level on the effective thermal conductivity of dry milk solids, the moisture content of the regular 1Manufactured by K. R. Wilson 00., Buffalo, New York. Model 37E, Ser.: 2576. 45 Figure 5.l.——Hydraulic Press = Hydraulic drum = Piston Cylinder and Anvil NW’UU II = Hydraulic pump = 1" Specimen - Insulation = 3" specimen 0013301 I = Cubic specimen Figure 5.2.——Specimen for Measurement of Solid Thermal Conductivity. 5" 1424 la: 4 re————— (a) For 1" specimen P ———>1 .- J 8 "““““ 3-3/8“"""" l ,. ._ ~42 - . I. / I P ,T I U I... 541 54' la - 8 / j / ’ l // . ,. . y/ i é/ (b) For3 spec1men ? : / 1 / I j .5)... /? ! 7 4 ' / / / im4 4 '/ 6,2.1/ = piston c = Cylinder / I gazéi SII specimen 4... l" __49 — ing co ar (c) For 5/8" cube specimen Figure 5.3.——Specimen Press. 47 powdered milk was raised to a higher moisture level before measuring Ke' This was done by spraying a thin layer of dry milk (1/2 in.‘or less) on 2 ft by 2 ft square pans and then keeping these pans in a high moisture room for several hours. The high moisture room is available in the dairy plant of Michigan State University with room temperature of 61.5°F and 100% relative humidity. Then the powder was kept for a few days in a sealed container for moisture equilibration. The moisture content of the powdered milk was tested by toluene distillation method (American Dry Milk Institute, Inc., 1965). Instron and Mechanical Properties of Particle Solid The mechanical properties of the dry milk solid were measured using an Instron testing machine1 available in the Agricultural Engineering Department (Figure 5.4). Figure 5.5 shows a sketch of the Instron System. The specimen was kept at position A of Figure 5.5(a). The cross-head moved downward at a constant speed, which could be adjusted. The balance gave the amount of loading so that by knowing the chart speed, the stress— strain relation of the specimen could be established. In addition, the deformations in both vertical and horizontal directions were measured by sensor C and D in Figure 5.5(a) and plotted by the X-Y Recorder after amplification. 1Manufactured by Instron 00., Canton, Mass. Model No. TM, Ser. No.: 1687. 48 Figure 5.4.——Instron Testing Facility C = Chart H = Cross—head TX = X—Transducer Q = Control S = Specimen TY = Y—Transducer Panel B = Balance XY = X—Y plotter X = X—sensor Y = Y-sensor P Operation Panel /’ \ ”" W to Y—Transducer .-... ”———‘-‘—‘———‘ 'Gear :Box: Chart g . Cross Head Full Scale Lgadlng oo 5 50 o oo o O E; H to m X- Zero Calibration m o© 0 oo ._, UGear Main y/r}t< : Box L o , (a) Instron: A = Specimen C = Y—sensor B = Cross Head D = X—sensor f Range 8 Range 0 O Selector O O Selector A “3° (00 \ ° @Q) 0 D O D X—Transducer Y—Transducer (b) Amplifier (c) X—Y Recorder Q) ..from f Q) \ufrom g Figure 5.5.——A Sketch of Instron System. 50 Transient Thermal Properties Measurement Facility The thermal conductivities of dry milk solids were measured using the facility shown in Figure 5.6. The major components of the facility are sketched in Figure 5.7. They are: l. Specimen Shell; which includes sample specimen and supporting block (A and B in Figure 5.7). 2. Insulation—Heater mechanism; the solenoids control the Heaters which swing in and out as necessary. 3. Upper Specimen Assembly (D in Figure 5.7); the copper block at the center acts as the heat source during the experiment. 4. Hydraulic System;1 the hydraulic cylinder C would raise the Specimen Shell and press the sample specimen against the heat source (U in Figure 5.7) during the experiment. 5. Control Panel;2 which controls the temperature of the heat source and the combination action of solenoids, heater and hydraulic pressure cylinder. 1Manufactured by Sperry Rand 00., Model No. CHJO 11609. 2Manufactured by Standard Electrical Prod. 00., Dayton, Ohio. 51 H = Heat Source S = Specimen Shell HY = Hydraulic Pressure Cylinder C = Control Panel P = Computer Signal Conditioner ll Heat Source S = Specimen in Position HY = Hydraulic Press Figure 5.6.——Optimun Transient Thermal Properties Measurement Facility. .spaaaosm meEmaSmmmz moflpamgoam HmEamQB pcoflmcth QSEHon go gopmxmll.m.m wadmfim Hosmm Hoapcoo O Empmzm xooap amgmoo aflflsmaomm momma .o .9 thEmmmm mmafloogw swamp nopcfifiho madmmosm oaazmhpzm .HHHHQ _ _ xooan mQHpMOszm / \ pmmp 90% :oEHoQO moccapfipcoo chmflm popsmsoo 52 3 _ _ w o oEmsm\\ Hamzm QmEHoQO awpmmm m. ampmom rs wUHocmHo )3 OOgI WEI 53 6. Computer Signal Conditioner;l which reads emf of each thermocouple and sends the temperature data directly to IBM 1800 Computer. Before conducting a test, the Computer Signal Con— ditioner, Control Panel and Hydraulic System had to be well balanced and adjusted. Then the sample specimen with thermocouple leads were put in position (Figure 5.6[b]). The heater was swung in to heat the copper block (U in Figure 5.7). As the test was started, the heater swung out by solenoids and, meanwhile, the sample specimen was raised by the hydraulic pressure cylinder (C in Figure 5.7) and eventually was pressed against the heat source, U in Figure 5.7. The temperature history of the sample specimen was punched out by IBM 1800 Computer. Figure 5.8 shows the dimensions of the specimen tested. The diameter of the copper block (U in Figure 5.6) is 3 in. The specimen shown in Figure 5.8(a) was prepared with the same diameter as the copper block and was tested without insulation around the side wall of the specimen. Computations in Appendix VI illustrate the amount of heat loss from the side wall. It is evident that the magni— tude of the heat loss from the side wall is about 7.4% of the heat transferred from the copper block. Therefore, insulation (styrofoam) was placed around the specimen of 1Manufactured by 8. Sterling 00., Application Divi— sion, Southfield, Michigan. 54 Q from hot copper 3/8 {L I "iii/16:: f al4:146: x2 —T—1/15 x -11}é16 v—I n X ‘fi an H 3/8 / «S X 4—1/8" l":@ x3 —*— X _.i_. 3 3/811 U2 0’ 4) £3 £1423 I / le____ 3..____ng (a) 3" diameter specimen (b) 3" diameter specimen Illa 1V“ , ,.. . J“0.0625" Y 1 2 3 0.5476" K x 1%" .l.‘ H —— — ——-————- (c) 1" diameter specimen Figure 5.8.——Thermocouple Locations of Specimens for Measuring Thermal Properties. 55 Figure 5.2 to eliminate the heat loss from the side wall. The whole facility was located in the Engineering Building of Michigan State University. SteadyrState Thermal Conductivity Measuring_Apparatus Figure 5.9(a) shows the layout of the entire apparatus. It was originally designed by tha et_al. (1966) and modi— fied by Farrall et_al. (1968). Powdered milk1 was heated by the electrical cable around the copper cylinder, Figure 5.9(b). The voltage of the heating cable was regulated depending on the desired temperature level of the powder for conducting the experiment. Usually it was around 42 volts or less to increase the mean powder temperature to around 140°F. Heat was transferred through the powdered material radially into the center tube where cooling water was flowing. The amount of heat transferred from the temperature rise and flow rate of cooling water was calculated. The constant head water device in Figure 5.9(a) gave a constant flow rate of cooling water which was kept at room temperature. The thermocouples were distributed across the radius and were held in position by a wooden holder (Figure 5.9[c]). 1Manufactured by Mid—west Producer's Creameries Inc., South Bend, Indiana. Extra Grade, Hi-Heat, Bakery Quality, Spray Process. 56 .msumamaq¢ wcHaSmmmz apfl>flpososoo HmEhoza mumpmuzpmmumll.m.m mhswfim Boos. «mad: 99522. udaooozxufi Nov 28:85. 3339!!» 29.553 0. Ham? 8 3:52.898 44:51» .6 29:23:53 63514.3( . «on. own: :2: 5.2.9528 :0: no 29.83 wmoxo A.3 9255902 >._._>.§ozoo .33:th uo .5053 AS baa! .1 I 5 4 DE! , 320.. .0 :30 .2 mums» / a; \s 4 l. a 2.... . $688255 «9. u 8 «IL; 623.0. .30.! was» qua-“fluid... “ . £253.80 3.. 23 5.093» or 5.0 .. a 1325.6 m =3; 4accsccco assuage caacm soc mpasmchI.:.e mamas 76 In addition, the computer predicted the effect of temperature on KS at 1.38 g/ml as shown in Table 6.5. Assuming a linear relation between temperature and Ks’ the results could be described by the following equation: Ks = 0.322 — 0.000376 T (6.8) The moisture content of the dry milk used in the test was 3.5% as determined by toluene distillation method. TABLE 6.5.—-Effect of Temperature on Ks' Tem erature 0 KS IEFI (g/ml) (Btu/(°F)(hr)(ft)) 75 1.38 0.27993 1.46 0.29400 200 1.38 0.23360 1.46 0.24700 Figure 6.7 shows the effect of temperature on Ks and Kg' Ks decreases slightly as temperature increases. Figure 6.8 shows the comparison of thermal con— ductivity of dry milk solids to other materials and agricultural products. Notice that thermal conductivities 77 0.30- 6.8) 0 0.25- .5 “ 0.20~— s.’ :E 0.15" s 4.) m 0.10-— U) m 0 05I— 0.00 I I I l J I l I I | l l I 75 100 150 200 Temperature, OF 0.0170 m 0| ,3 0.0160 m :4 g '\ 3 o 0150 m . "do _ M 75 100 0 150 200 Temperature, F Figure 6.7.——Effect of Temperature on Thermal Con— ductivity of Milk Solids and Air. 1.30 - OF .90 — .80 — ft. .70-— .60 — Ks’ Btu/hr. .20I— .10 —— 78 'Marble 'Ice Fésh Muscle a Clay . Concrete Milk Solids 'Wood Asbestos Diatomaceous Earth I I I I I I I I I .00 —40 Figure 6 ductivities of —20 0 20 40 60 80 100 Temperature, OF .8.—-Temperature Effect on Thermal Con— Food Products and other Material. 200 79 of all food products decrease as temperature increases (Woodams and Nowrey, 1968). Effective Thermal Conductivities of Dry Milk Effective thermal conductivities of several types of powdered milk at various levels of bulk densities and moisture content were investigated by Farrall et a1. (1968). The values of Ke were calculated by the equation as follows (Deissler and Boegli, 1958): R (i) Ke = Q loge R1 2 fl 1 (T2 — Tl) where T1’ T2 are the temperatures at radii R and R l 2’ respectively, 8 is the length of cylinder and Q is the heat flux transferred from the heating coil to the water flow at the center of the cylinder (Figure 5.9). Table 6.6 shows the results for Ke of nonfat dry milk. Figure 6.9 shows the effect of moisture content on the Ke of nonfat dry milk. It was correlated as follows: Ke = 0.114 + 0.0107 MC (6.9) by Mathatron 4280 computer with coefficient of correlation 0.95, where M0 is the moisture content in percentage. Btu/hr. ft. OF Ke’ 80 .28 — I I I I .05 I I IIIIIIIIJIIL 4 5 6 7 8 9 10 11 12 Moisture content, % O }._4 N LA) Figure 6.9.——Effect of Moisture Content on Ke of Nonfat Dry Milk. 81 The average temperature of the bed for experiments to measure effective thermal conductivities was shown in Table 6.6. TABLE 6.6.--Effective Thermal Conductivity of Regular Non- fat Dry Milk.* Moisture, % Mean Temperature, Bulk Density, e °F (g/ml) 2.2 0.1391 144 0.594 3.4 0.1331 137 0.564 4.5 0.1430 140.4 0.605 4.6 0.1263 153 0.575 3.4 0.1331 137 0.564 4.0 0.1364 138 0.563 7.8 0.1966 151 0.533 7.9 0.1976 116 0.547 10.0 0.2311 125.7 0.588 10.4 0.2242 129.5 0.422 11.9 0.2327 118.6 0.544 * From Farrall et a1. (1968). Number of Contact Points of Spheres in a Random Bed In Equation (A31) the parameter n is half of the average number of contact points of uniform spheres in a packed bed. Kunii and Smith (1960) obtained the following equation to estimate the parameter n: 82 n = 6.93 - 5.51 M (6.10) which gives n = 1.42 for the most open packing (a = 0.4760) of spheres and n = 6.93 for the most dense packing (e = 0.260). The agreement between experimental data and predicted values for 2 n in Table 6.7 suggests that Equation (6.10) might be valid for prediction of contact number of uniform spheres in a random bed from the void of the bed. 83 mm\ma\m ma m:.m mmmz. ma.s =m mw\ma\m :m mm.m sman. zo.s :m.H mm\: \m mm m:.m mmmq. ms.m :m mama oopcsoo monondm Aoa.wv coapmsum Hmpcoeflmodxm moLoSQm go 909552 Eonm .cm hpfimonom «Cm mo mNHm .oom Eoocmm m CH mmpondm uo mpcflom homecoo mo thESZII.>.m mqm e s 1F x=xmax - g(x) dX] (A7) X=S where M1F and integration of g(x) were evaluated by Romberg numerical integration methods (Moursund and Duris, 1967). Appendix II and III shows the computer program used to evaluate M and integration of g(x) respectively. 1F For regular nonfat dry milk, x0 = 45.5u = 0.1493 x 10'3 ft, xmin = 5u = 0.164 x 10'“ ft, (7.10) 9O 180u = 0.59 x 10'3 ft, >4 II 1.67, Q II the results of integration were: _ -9 MlF - 35.22 X 10 X max g(x) dx = —l.255 (7.11) Xmin Substituting Equation (7.10)and Equation (7.11) into Equation (A7), a simplified expression of G(s) for this particular particle size and size distribution of regular nonfat dry milk was obtained as following: (1802 - s2) __—_180__—— a G(s) = 1.45 s [0.027 + 1.255] (7.12) Equation (7.12) was plotted as shown in Figure 7.1. By varying x0 and o, the effect of x0 and c on G(s) is illustrated. Since the frequency distribution function G(s) should be a good approximation of the number of contact points on the cross—section as s + 0+, the values of G(s) were cal— culated as 8: 0.500 for different combination of x0 and 0. Table 7.1 shows that the smaller particles and lower stand- ard deviation (uniform particles) would have a larger number of contact points expected on the cross—section. in NIH? , . . . . II II..I....,...._.\W(I LII in: .Iob. u. 44. IIII 91 o = 1.67 -—-— 0 = 1.475 440 .— / A = Regular nonfat dry milk B = Instant nonfat dry milk .// 400 —- / / / 360 — / / G(s) 0 30 60 90 120 150 180 S. u Figure 7.1.—-Effects of Mean Particle Size and Standard Deviation of Size Distribution on Function G(s). 92 TABLE 7.1.——Predicted Number of Contact Points on the Cross—Section of a Random Bed (11 73).* Mean Particle Size, x Standard 0 DeViation, U “5.5“ 7511 12211 1.67 0.930 0.396 0.206 1.475 1.220 0.468 0.252 *Roman numbers refer to research notebook number; Arabic number refers to research notebook page number. 3. Contact Conductance By knowing the contact conductance of total con— tact points,KC, and the number of contact points on the cross—section, the contribution of the thermal conduc— tance through a single contact point to Ke could be deter— mined from the equation: K345 = Kc/G(0.50u) where Kc = 0.00292 BtU/(hr)(°F)(ft) G(O.50u) = 0.930 from Table 7.1 for regular nonfat dry milk. Thus K345 = 0.00314 Btu/(hr)(°F)(ft) (4.26) (7.9) (7.13) film. h. I. 93 which is negligible when compared to the Ke value in Table 6.6. However, since K0 is a function of n, 8, x0, h T and MC by Equation (4.18) and G(0.1u) is K K PS’ 8’ g: a function of x o and s by Equation (4.12), o’ Xmin’ Xmax’ the effect of these parameters on contact conductance, K345’ can be investigated independently for academic interest using the methematical model developed. Effect of Each Parameter on Kn There are eight basic parameters involved: E, v, o, N, T, MC and e The solid thermal conductivity, F' K was expressed in terms of T and MC in Equation (7.8) for regular nonfat dry milk. Gas thermal conductivity, K can be expressed in terms of T as Equation (4.25) for air. Since the value of Ke is additive as shown in Equa— tion (4.27), the effect of each parameter on Ke can be analyzed by the effect of that parameter on each components: K1, K2 OT K345: In the previous section it has been shown that E, v, x o, and N influence K345 only, which, in turn, has 0, a negligible influence on Ke‘ Therefore, the effects of E, 0, x0, 0 and N on Ke are negligible. .7! 94 Equation (4.27) is then reduced as follows: K = Kl + K2 (7.14) where K1 and K2 are defined by Equation (4.21) and Equation (4.23) respectively. By Equation (7.8) and Equation (4.25): K = (0.2851 - 0.000376 T + 0.0107 MC) (1 — 8F) (7.15) K2 = (0.0133 + 0.0000 21 T) 6F (7.16) Figures 7.2 through 7.4 show the effect of temperature, moisturecontent and bulk density, respectively, on K1’ K2 and K8. Notice that in all cases K1 is the dominant factor as far as the value of Ke is concerned. Table 7.2 shows the percentage contributions of the values of K1’ K2 and K345 to Ke° By average, IX ll 1 0.933 Ke K = 0.048 Ke (7.17) 345 = 0.019 Ke claw... , . . . :w: .. ..-.I.I-III, OF Thermal conductivity, Btu/hr.ft. .30 .25 .20 .15 .10 .05 .00 95 e = 0.51 ‘ MC = 3.5% Temperature, OF Figure 7.2.——Effect of Temperature on the Component and Effective Thermal Conductivities. ..._..._ ..I.I.b.. {Manly IAIJELAWHEEEII .30 OF .25 .20 .10 Termal conductivity, Btu/hr. ft. .05 96 e = 0.51 = 140°F I era I- _. K2 ____. ______.__11__1/_________. I I I I, I I I I I I I L 0 2 4 8 10 12 Moisture content, % Figure 7.3.—-Effect of Moisture Content on the Com- ponent and Effective Thermal Conductivities. 97 —— T = 140°F —— MC = 3.5% g/ OF ft. Ke’ Btu/hr. .05 — —- K __L__2 "/,I_-'T"'T 1_‘-tT'—j—--+-—-L_ I .5 I 1.0 1.5 p. s/ml Figure 7.4.—-Effect of Bulk Density on the Com— ponent and Effective Thermal Conductivities. 98 The experimental values of Ke are from Table 6.6 and Table 6.4. The agreement between prediction and experi- mental values is illustrated. Figure 7.5 shows the com— bined effect of temperature and moisture content on Ke of nonfat dry milk. Since K1 is the dominating component of Ke’ Equation (4.21) implies that KS and e are the dominating F factors as far as Ke value is concerned. Since KS could be expressed in terms of T and MC as shown in Equation (7.8), therefore the dominating parameters of Ke values of regular nonfat dry milk are T, MC and e F. Since K2 is the minor component of Ke, the effect of Kg on Ke is minor as shown in Figure 7.6 Due to this fact, the different kinds of interstitial gasses will have only limited influence on the Ke value of nonfat dry milk. TABLE 7.2.——Per cent of K1, K2 and KC to Ke (5F = 0.51) (II 84). Temp., °F MC, % Kl, % K2, % KC, 7 75 3.5 93.6 4.4 2.0 140 3.5 91.8 6.3 1.9 180 3.5 91.5 6.4 2.1 75 8.0 94.1 4. 1.9 140 8.0 93.5 6.4 1.9 180 8.0 92.8 5. 2.0 75 12.0 94.9 3.3 1.8 140 12.0 94.2 3.9 1.9 180 12.0 93.5 4.6 1.9 AVERAGE: KO U) W .t‘ CD |_I KO I \I..1II I .30 .25 OF .20 ft. .15 Btu/hr. Ke’ .10 .05 .00 99 __ Al2% " CI 8% — 03.5% - A _~ A : 777777777 1-313%;140 - CI [3 7‘ \ " \ \ \ \ 8% MC __ O ____________—-_—_—- __ C) (}{)___()__;:£E%J£C — O _ O -. I I I I I I I I I J I 100 140 180 Temperature, OF Figure 7.5.-—Effect of Temperature and Moisture Con— tent on Effective Thermal Conductivity. al- 01. 100 .30 — T = 14OOF _ Ef=0.5l I— .25 - .12 - m _. o ._ g I C) Q .15 _f1 (3 '5: 1:1 ‘3 -— 002 5; J.) m __ “ —' Air 9 10 —— M u __ N2 .05 '- .00 I I I I I .011 .012 .013 .014 .015 .016 Kg, Btu/hr. ft. OF Figure 7.6.-—Effect of the Thermal Conductivities of Interstitial Gases on the Effective Thermal Conductivity of Nonfat Dry Milk. all 101 Influence of Component Thermal Conductivities on Effective Thermal ConductiVity Using Dimensionless Groups In order for the mathematical models developed to have broad application on various kinds of solid material and interstitial gases, two dimensionless groups, Ks/Kg and Ke/Kg’ were plotted as shown in Figure 7.7 as a function of void fraction, 6. For this particular case (nonfat dry milk), the effect of moisture content at one temperature level, 75°F, is illustrated. For a given combination of Ks’ Kg and s, Figure 7.7 shows that Ke will be higher for higher moisture content as predicted. Comparison of Predicted and Experimental Thermal Conductivities Figure 7.8 shows the comparison between the predicted and experimental values of K8. The mathematical model developed to predict the effective thermal conductivities of organic powder in a packed bed was verified by the agreement in Figure 7.8. The experimental data were correlated by Mathatron 4280 computer. The result was as follows: log (Ke) = 0.3480 + 1.4051 log (Ke) observed predicted (7.18) with coefficient of correlation equaled to 0.96. 102 1000 — e=0.%3 100:: ml 60 M x T 10:: __ .____ 12% MC — ——-——— 8% MC _ —— 3.5% MG I I I I II III I I I I II III 5 10 50 100 K /K e E Figure 7.7.-—Correlation for Thermal Conductivity of Nonfat Dry Milk in Packed Bed. 103 .3 : m I O .25 _ Observed.>/ S I {é 9 _ // Predicted A _ s .2 - / g C9 p _ / m / . ' / " - / I) g .15 -— <3 / £5 — ’3 a ‘ /8 m / x: _ .1 — I I I I I I I I II II III LIIIII .10 .15 .2 .25 .3 Ke (Predicted), Btu/hr. ft. OF Figure 7.8.——Comparison of Predicted and Experi— mental Ke for Nonfat Dry Milk with Air as Interstitial Gas. CHAPTER VIII CONCLUSIONS 1. The primary mechanism of heat transfer through nonfat dry milk in a packed bed was found to be conduction. Convection and radiation were found to be negligible. 2. Heat conduction through nonfat dry milk in a packed bed was divided into three components in parallel, i.e., through particle solid, through interstitial gas and through contact points between particles. The contribution of the solid phase was found to be the dominating factor. The contribution of the gas phase was found to be a minor factor and that through contact points was found to be negligible. 3. A mathematical model was developed to predict the effective thermal conductivity of nonfat dry milk in a packed bed based on the three mechanisms mentioned above. The model was verified by comparing the predicted and the experimental data. 4. There are eight basic parameters considered in developing the mathematical model: mean particle size, particle size distribution, modulus of elasticity, Poisson's ratio, number of contact points, temperature level, moisture content and void fraction. The effects of mean particle 104 105 size, size distribution, modulus of elasticity, Poisson's ratio and number of contact points were found negligible because the contact conductance was negligible. 5. The number of contact points can be predicted from the porosity of the packed bed. 6. The effects of temperature, moisture content and void were significant because the solid thermal conductivity was the dominating factor. 7. The contribution of thermal conductivity of the interstitial gas to the effective thermal conductivity of the powder bed was found to be a minor factor. Different types of interstitial gases should not have a significant influence on the values of effective thermal conductivity of a powder bed. 8. The number of contact points appearing on the cross—section of a packed random bed will increase as particle size and standard deviation of particle size decreases. 9. By the relation established between bulk density, void and pressure, the distribution of the effective thermal conductivities of an organic powder along the vertical axis of a large container (silo, etc.) can be predicted, based on the mathematical model developed. CHAPTER IX LIMITATION OF THE MATHEMATICAL MODEL DEVELOPED The mathematical model presented in Chapter IV was developed to predict the effective thermal conductivity of an organic powder in a packed bed. Since five assumptions were made in the development, the model has the following limitations: 1. The mean particle size must be less than one millimeter and temperature must be lower than 400°C in order to neglect the mechanism of radiation. 2. The mean particle size must be less than 3 m 4 millimeter and the gas pressure must be lower than 8 m 10 atmospheres in order to neglect the mechanism of convec— tion. 3. The solid material of the powder must have a low thermal conductivity so that the assumption of one dimensional heat flow is valid. 106 CHAPTER X RECOMMENDATIONS FOR FUTURE WORK 1. Determine the effect of moisture content on the thermal conductivity of dry milk solid experimentally. 2. Measure the Poisson's ratio by bulk modulus as suggested in Chapter VI. 3. Conduct an independent study on the effect of particle size and size distribution, modulus of elas- ticity, Poisson's ratio and contact number on the contact conductance. 4. Measure contact conductance for a packed bed of spheres with known solid thermal conductivity under vacuum and compare the value with the prediction by the mathe— matical model developed. 5. Verify the assumption 5 in Chapter IV by measur— ing the thermal conductivity of talc powder compressed to the same bulk density as talc block, and then compare the results with the thermal conductivity of the solid. 6. Measure the solid thermal conductivity of non— fat dry milk in the low temperature range to obtain the ef— fect of temperature on solids thermal conductivity experi— mentally. This could be done by cooling the specimen by dry ice instead of heating the specimen by hot copper block as was done in this investigation. 107 BIBLIOGRAPHY American Dry Milk Institute, Inc. 1965. Chicago, Illinois. Standards for Grade of Dry Milk Including Method of Analysis. Bulletin 916 (Revised). 51 pp. Argo, W. B. and J. M. Smith. 1953. Heat transfer in packed beds. Chem. Eng. Progress 49:443—451. 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APPENDIX I REDUCTION OF EQUATION (4.12) max G(s) = $- ———FLfl2—/— dx (14.12) 1F (x - s )2 x=s 1n E -».< °>2 F(x) = W e 1n 0 V2n 1n 0g (4.10) x c2 (mfg)2 = C [e l where C]. = _____L_ C2 = _ __l__2 , (Al) V2n 1n 0, 2(1n o) _ g 2 013 x_xmax [602 (1“ I3 1 G(s) = -— —"‘2—2£' dx (A2) 1F (x — s )2 Notice that at the low limit of Equation (A2) function is not defined. The technique of integration by parts could be applied to Equation (A2) as follows: 116 117 2 2)+% (A3) Let u = % e o and V = (x — s then _» dV = x(x2 - $2) 2 (A4) and 2 x 2 C (1n i) c (1n = _ L _g 2 X0 2 X0 (-l) du — x dx (e ) + e 2 x 2 x x 2 C 1n = C (1n =) _ _l_ 2 X . E 1 _ l__ 2 X0 _ X e O 2C2 (1n X)(X) 2 e o x 2 x 2 x 20 C 1n = C 1n = = ——% (1n i) ‘ e 2 X0— £5 e 2 x0 x x0 x C2 ln2 i- e X0 — (2C2 1n = 1) —————;§*——— (A5) Apply Equation (A3) and (A5) to Equation (A2) as follows: X C S max g(s) = ML. 1F u dV x x C S max max = l [uv - V du] 1F s s C S C 2 x Xmax X=xmax L 1n = — = ML [31? (x2_s2)2.e 2 X0 _ 1F s x-s 2 1 c 1n2 % 2 / (x—s>2(2c21n§=_1)e 2 de3 (A6) X 1 x=x c S 2 x xmax max L C 1n = — = Mi[%(x2—s2)2 e 2 x0 8 1F x=s g(X) dx] (A7) where . C2 1n2 %b L g(x) = (x2 — $2)2 (2 c2 1n = 1) §———2———— (A8) x x max M1F = x ' F(x) dx (4.13) Xmin x max 02 (1n %)2 = X Cl e 0 dx (A9) x 119 Equation (A8) could be integrated by numerical integration as Appendix II. Also Appendix III shows the computer pro— gram for evaluating Equation (A9). 120 APPENDIX II COMPUTER PROGRAM FOR INTEGRATION OF EQ. (A8) 'FOR.L9X H V 10 11 PRQGRAM ROMB§RG DIMENSION 9(11011) PEAD.19$QXMAXJK FORMAT (2F10-0912) pRINT,2xSOXMA54K FORMAT (2E20010o13) H=XMAX?S P(191)=05*H*(F(S)+F(XMAX)) KP=K+1 KC=1 004 I=ltK V=Oo DO 3_J=10KC X=J V=V+F(S+(X-q5)*H) V=V*H F’( 1+1 9 1 1315* ( P (J. .9.LL10:M) KC=2*KC H=c5%H W=4o DO 8 I=21KP WM=W~10 DO 6 J=IQKP p(JOI)=(W*P(JQI-l)-P(J-loI-l))/WM W=40*W DO 10 I=19KP PRINT 119(P(19J)0J=L~I) FORMAT (11E12o3) END FUNCTION F(X) S:500 C2=-1o9 XMEAN=4505 XCENT=X/XMEAN T=LOGF(XCENT) U=T**Z F=SQRT(X**2-5**2)*(2*C2*T-1)*ExP(cg*U)/x**2 RETURN END 'RUN.O.45~2100 500 180.010 121 APPENDIX III COMPUTER PROGRAM FOR EQ. (A9) 'FOQ9X9L O O p.- RROGRAM GAU MlF CALL CADSSN¢~I.5.o.180.o.v.1.0E—6.Io> PRINT 1.v FORMAT<1HI.EI4.7) END SUEROUTINE GAUSSN(INIToXOoXLchRELoNP) TO CONVERT FROM GAUSSlé T0 CAUSSN. CHANGE THE CARDS WITH COMMENTS. WHERE N = ORDER OF FORMULA. DIMENSION AAIIe).HH(IE).YBAR<10).BYB